Local sharp maximal functions
The authors consider in detail the propserties of the local maximal function introduced by F. John namely, for \(0<\alpha \leq\) set \[ M^{\#}_{0,\alpha}f(x)=\sup_{x\in Q}\inf_{c} \inf \{A\geq 0: | \{y\in Q: | \quad f(y)-c| <A\}| <\alpha | Q| \}, \] where \(c\in {\mathbb{C}}\) and Q denotes a cube in \({\mathbb{R}}^ n\) of sides parallel to the coordinate axes. Among other things they show that the Peetre K-functional for real interpolation between the Lorentz space \(L^{1,\infty}({\mathbb{R}}^ n)\) and BMO(\({\mathbb{R}}^ n)\) satisfies \[ K(t,f;L^{1,\infty}({\mathbb{R}}^ n),BMO({\mathbb{R}}^ n))\approx \sup_{0<s<t}s(M^{\#}_{0,\alpha}f)^*(s), \] where * denotes, as usual, the non-increasing rearrangement of f. Several applications of this careful study are given.
scientific article; zbMATH DE number 3901395
scientific article